Some friends and I are having trouble with a functional equation problem :
If $f : (0,1) \to \mathbb R$ is a positive continuous function satisfying $$\int_t^1 f(x)f\left(\frac{t}{x}\right)dx = \sqrt{t}$$ for all $t$, then $f(x) = \sqrt{\frac{2x}{\pi(1-x^2)}}$.
The function is not in $L^2$ or any nice Hilbert / normed space, so functional analysis techniques seem out of reach. I've tried various changes of variables, like $y = x /\sqrt t$ and $y = t/x$, but none of these yielded anything valuable. I've also tried integrating over u, but it didn't do much.
The problem might reduce to proving that two solutions $f,g$ have to be equal almost everywhere / on a dense subset of $(0,1)$, as we've been able to show that the given function is indeed a solution.
Any ideas would be welcome !